Two men on either side of a pole of 30 m high observe its top at angles of elevation α and β respectively. The distance between the two men is 40√3 m and the distance between the first man at A and the pole is 30√3m.

Based on the above information, answer the question:

tan(α + β) =

Infinite (does not exist)

$tan β =\frac{BD}{AD}$

$tan β =\frac{30}{10\sqrt{3}} \Rightarrow tan β =\frac{3}{\sqrt{3}}\Rightarrow tanβ=\sqrt{3}$

$β =tan^{-1}(\sqrt{3})$

$β =60°=\frac{\pi}{3}$ .....(ii)

From eq. (i) & (ii)

$tan(\frac{\pi}{6}+\frac{\pi}{3×2})=tan(\frac{π+2π}{6})=tan(\frac{3π}{6})=tan\frac{π}{2}$

Infinite (it does not exist)

So, option 4 is correct.